atom
Let be a poset, partially ordered by . An element is called an atom if it covers some minimal element of . As a result, an atom is never minimal. A poset is called atomic if for every element that is not minimal has an atom such that .
Examples.
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1.
Let be a set and its power set. is a poset ordered by with a unique minimal element . Thus, all singleton subsets of are atoms in .
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2.
is partially ordered if we define to mean that . Then is a minimal element and any prime number is an atom.
Remark. Given a lattice with underlying poset , an element is called an atom (of ) if it is an atom in . A lattice is a called an atomic lattice if its underlying poset is atomic. An atomistic lattice is an atomic lattice such that each element that is not minimal is a join of atoms. If is an atom in a semimodular lattice , and if is not under , then is an atom in any interval lattice where .
Examples.
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1.
, with the usual intersection and union as the lattice operations meet and join, is atomistic: every subset of is the union of all the singleton subsets of .
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2.
, partially ordered as above, with lattice binary operations defined by , and , is a lattice that is atomic, as we have seen earlier. But it is not atomistic: is not a join of ’s; is not a join of and are just two counterexamples.
Title | atom |
---|---|
Canonical name | Atom |
Date of creation | 2013-03-22 15:20:09 |
Last modified on | 2013-03-22 15:20:09 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 13 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06A06 |
Classification | msc 06B99 |
Defines | atomic poset |
Defines | atomic lattice |
Defines | atomistic lattice |
Defines | atomistic |